In the simple case of equal probabilities, this entropic functional is given by \(S_{BG}=k \ln W\) . Eq. (REF ) is generically additive [1]}. Indeed, if \(A\)  and \(B\)  are two probabilistically independent systems (i.e., \(p_{ij}^{A+B}= p_i^A p_j^B\) ), we straightforwardly verify that \(S_{BG}(A+B)=S_{BG}(A)+S_{BG}(B)\) . This celebrated entropic functional is consistent with thermodynamics for all systems whose \(N\)  elements are either independent or weakly interacting in the sense that only basically local (in space/time) interactions are involved. For example, if we have equal probabilities and the system is such that the number of accessible microscopic configurations is given by \(W(N) \propto \mu ^N\; (\mu >1; \,N\rightarrow \infty )\) , then \(S_{BG}(N)\)  is extensive as required by thermodynamics.
Indeed \(S_{BG}(N)=k\ln W(N) \sim k(\ln \mu )N\) . But if the correlations are nonlocal in space/time, \(S_{BG}\)  may become thermodynamically inadequate. Such is the case of equal probabilities with say \(W(N) \propto N^\nu \;(\nu >0; \,N\rightarrow \infty )\) : it immediately follows \(S_{BG}(N) \propto \ln N\) , which violates thermodynamical extensivity.
To satisfactorily approach cases such as this one, it was proposed in 1988 [2]}, [3]}, [4]}, [5]} to build a more general statistical mechanics based on the nonadditive entropic functional
\(S_q\equiv k\frac{1-\sum _{i=1}^W p_i^q}{q-1}=k\sum _{i=1}^W p_i \ln _q \frac{1}{p_i} = -k\sum _{i=1}^W p_i^q \ln _q p_i = -k\sum _{i=1}^W p_i \ln _{2-q} p_i \;\;(q \in \mathbb {R}; S_1=S_{BG})\,,\) 
